Systems and methods for determining reflection and transmission coefficients

ABSTRACT

A method is provided for calibrating a terminal device connected to a transmission line containing an impairment. The method includes steps of obtaining a sequence of frequency domain samples for a digital signal transmitted to the terminal device, determining a reflection coefficient from the obtained frequency domain sequence and a reflection signal arising from the impairment, converting the sequence of frequency domain samples and the frequency domain reflection signal into the time domain to generate a complex time domain sample sequence having a real I time component and an imaginary Q time component, correcting the time domain sample sequence into a corrected time sequence having a phase value of the Q component corresponding to a phase value of the I component, calculating a correcting spin coefficient from the corrected time sequence, and calibrating the terminal device with the correcting spin coefficient to mitigate a rotation of the reflection coefficient.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of and priority to U.S. ProvisionalPatent Application Ser. No. 63/032,987, filed Jun. 1, 2020, thedisclosure of which is herein incorporated by reference in its entirety.

BACKGROUND

The field of the disclosure relates generally to digital transmissionsystems, and more particularly, to multi-carrier wired, wireless, andoptical digital transmission systems.

Conventional digital transmission systems often exhibit both linear andnon-linear distortion. Conventional digital transmission systems utilizesymbols with coefficients, either in the time domain (TD) or frequencydomain (FD), which are generally complex-value sequences. That is, thecoefficients of the complex symbols typically include both a real (Re)component and an imaginary (Im) component, or alternatively, a magnitudeand a phase value. The time and frequency domains of the transmissionsystems are related and, for a plot or a sequence of numerical values,it must be known whether to observe the plotted numbers as time domainor frequency domain values. This distinction is of particularsignificance when considering multi-carrier (MC) digital transmissions,such as with orthogonal frequency division multiplexing (OFDM) andorthogonal frequency division multiple access (OFDMA) transmissions.

OFDM symbols, for example, when plotted, appear as discrete values inthe frequency domain, but look more like random noise in the timedomain. In contrast, if a transmission is single carrier (SC), itssymbols can be viewed as discrete values in the time domain, but looklike random noise in the frequency domain. Multi-carrier and singlecarrier transmissions are thus typically viewed in different domains.

One type of interference/distortion that severely affects digitaltransmissions is multipath linear distortion, which is sometimesreferred to as “reflections,” “echoes,” “ghosts,” or “dispersion.” Anexample of such distortion occurs when a data transmission (e.g., abaseband radio frequency (RF) signal) is sent over a direct path betweena transmitter and a receiver, but is also reflected off at least oneobject or impedance mismatch outside of the direct path. In such cases,the receiver receives the main signal of the direct path, but also areflection of the signal over the indirect reflection path. Suchreflections combine with the main signal over the direct path, therebycausing distortion in the received signal.

On wired signal paths, reflections may also occur from impedancemismatches within coaxial networks, such as in the case where one ormore copies of the original signal, which may include a delay and/or anattenuation, are added to the original signal. In comparison, onwireless signal paths, multipath linear distortion may arise fromsignals reflected off of physical structures.

Where the wired signal path includes single mode fiber optic coherentoptical signals, on the other hand, the transmission characteristicsdiffer from RF wired and wireless signal paths. That is, for a singlemode glass fiber, impairments such as Chromatic Dispersion (CD), whichis similar to group delay, may occur at lower frequencies, since signalsat different frequencies (e.g., optical wavelengths) travel at differentspeeds down the fiber optic cable, and the linear distortion therefromis typically equalized to minimize inter-symbol interference (ISI). Thisimpairment becomes more pronounced with longer fiber optic cable spansand increasing bandwidth, and is different from echoes, which are nottypically encountered on fiber optic transport media. Some distortionequalization systems and methods are described in greater detail inco-pending U.S. patent application Ser. No. 16/927,802, filed Jul. 13,2020, the subject matter and disclosure thereof which is incorporated byreference herein. The following description may broadly refers to suchdistortions and impairments collectively as “interference(s).”

More particularly, as most easily seen on a Smith Chart, the F magnitudeof a reflection coefficient of a reflection may remain constant over alength of a transmission line, but the F phase of reflection coefficientwill change over that same length. On a conventional Smith Chart (i.e.,a standard tool for analysis of transmission lines, filters, andmatching circuits used to plot the reflection coefficient F againstfrequency in a complex plane), this reflection coefficient will be seenas a constant-radius rotation, or spin from a point representing a loadvalue for F. The longer the length of cable in question, the faster thespin on reflection coefficient. To compensate for this rotation,conventional techniques require a technician to perform an on-sitecalibration procedure at the modem using a specified, known length ofdrop cable (e.g., 100 ft) connected to the modem. It is thereforedesirable to be able to calibrate the downstream device withoutrequiring a costly on-site technician visit.

In the case of a cable system, the length L_(D) is unknown and thuscannot be corrected for the modem without such on-site drop-cableprocedures. Thus, the angle, or spin, of the reflection coefficientcannot be determined remotely in the conventional system. Knowing theangle of the reflection coefficient provides useful knowledge to thenetwork operator, not only with regard to the location of damage to thetransmission line, but also regarding the nature of such damage. Forexample, an open circuit measurement (hereinafter, “an open”) mayindicate that possibly a coaxial shield has been broken, a terminationscrew became or was left untightened, or a center pin was not makingcontact in a connector. In contrast, a short circuit measurement(hereinafter, “a short”) might indicate that a cable has possibly beencrushed, thereby pushing the shield into contact with the centerconductor.

It is therefore further desirable to able to determine, from capturedspectrum data on the transmission line, both the magnitude and phase ofthe reflection coefficient from various particular points along thetransmission path so that both the location and the nature of damage tothe transmission line may be easily determined without costly on-sitetechnician testing.

BRIEF SUMMARY

In an embodiment, a method is provided for calibrating a terminal deviceconnected to a transmission line containing at least one impairment. Themethod includes a step of obtaining a sequence of frequency domainsamples for a digital signal transmitted to the terminal device over thetransmission line. The method further includes a step of determining,from the obtained frequency domain sequence, a reflection coefficientfrom a frequency domain reflection signal of the digital signal arisingfrom the at least one impairment. The method further includes a step ofconverting the sequence of frequency domain samples and the frequencydomain reflection signal into the time domain to generate a complex timedomain sample sequence having a real I time component and an imaginary Qtime component. The method further includes a step of correcting thetime domain sample sequence into a corrected time sequence having aphase value of the Q time component corresponding to a phase value ofthe I time component. The method further includes steps of calculating acorrecting spin coefficient from the corrected time sequence, andcalibrating the terminal device with the spin coefficient to mitigate arotation of the reflection coefficient.

In an embodiment, a network analyzer is coupled with a transmission lineof a digital communication network. The network analyzer includes aprocessor in operable communication with the transmission line, and amemory device for storing computer-executable instructions. Theinstructions, when executed by the processor, cause the processor toobtain a sequence of frequency domain samples for a digital signaltransmitted over the transmission line to a downstream terminal device,and determine, from the obtained frequency domain sequence, a reflectioncoefficient of a frequency domain reflection signal arising from aphysical impairment of the transmission line disposed between thenetwork analyzer and the downstream terminal device. The instructionsfurther cause the processor to convert the sequence of frequency domainsamples and the frequency domain reflection signal into the time domainto generate a complex time domain sample sequence having a real Icomponent and an imaginary Q component in the time domain, calculate acorrecting spin coefficient from the corrected time sequence, andinstruct the downstream terminal device calibrate a rotation of thereflection coefficient arising from the physical impairment using thecalculated correcting spin coefficient.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the presentdisclosure will become better understood when the following detaileddescription is read with reference to the following accompanyingdrawings, in which like characters represent like parts throughout thedrawings.

FIG. 1 is a schematic illustration depicting a wired transmissionsystem.

FIG. 2 is a graphical illustration depicting a Smith Chart plot of theparticular energy recursion depicted in FIG. 1 .

FIG. 3 is a schematic illustration depicting an exemplary wiredtransmission system, in accordance with an embodiment.

FIGS. 4A-B are graphical illustrations depicting Smith Chart plots forthe transmission and reflection signals, respectively, measured by thesystem depicted in FIG. 3 .

FIG. 5 is a graphical illustration depicting a comparative plot ofsimulated frequency spectral data captured by the network analyzerdepicted in FIG. 3 .

FIG. 6 is a graphical illustration depicting a comparative time domainplot of the comparative frequency plot depicted in FIG. 5 .

FIG. 7 is a is a graphical illustration depicting a phase correctionplot of the comparative time domain plot depicted in FIG. 6 .

FIG. 8 is a graphical illustration depicting a conversion plot of thephase correction plot depicted in FIG. 7 .

Unless otherwise indicated, the drawings provided herein are meant toillustrate features of embodiments of this disclosure. These featuresare believed to be applicable in a wide variety of systems including oneor more embodiments of this disclosure. As such, the drawings are notmeant to include all conventional features known by those of ordinaryskill in the art to be required for the practice of the embodimentsdisclosed herein.

DETAILED DESCRIPTION

In the following specification and the claims, reference will be made toa number of terms, which shall be defined to have the followingmeanings.

The singular forms “a,” “an,” and “the” include plural references unlessthe context clearly dictates otherwise.

“Optional” or “optionally” means that the subsequently described eventor circumstance may or may not occur, and that the description includesinstances where the event occurs and instances where it does not.

Approximating language, as used herein throughout the specification andclaims, may be applied to modify any quantitative representation thatcould permissibly vary without resulting in a change in the basicfunction to which it is related. Accordingly, a value modified by a termor terms, such as “about,” “approximately,” and “substantially,” are notto be limited to the precise value specified. In at least someinstances, the approximating language may correspond to the precision ofan instrument for measuring the value. Here and throughout thespecification and claims, range limitations may be combined and/orinterchanged; such ranges are identified and include all the sub-rangescontained therein unless context or language indicates otherwise.

As used herein, the terms “processor” and “computer” and related terms,e.g., “processing device”, “computing device”, and “controller” are notlimited to just those integrated circuits referred to in the art as acomputer, but may also broadly refer to a microcontroller, amicrocomputer, a programmable logic controller (PLC), an applicationspecific integrated circuit (ASIC), and other programmable circuits, andthese terms are used interchangeably herein. In the embodimentsdescribed herein, memory may include, but is not limited to, acomputer-readable medium, such as a random access memory (RAM), and acomputer-readable non-volatile medium, such as flash memory.Alternatively, a floppy disk, a compact disc-read only memory (CD-ROM),a magneto-optical disk (MOD), and/or a digital versatile disc (DVD) mayalso be used. Also, in the embodiments described herein, additionalinput channels may be, but are not limited to, computer peripheralsassociated with an operator interface such as a mouse and a keyboard.Alternatively, other computer peripherals may also be used that mayinclude, for example, but not be limited to, a scanner. Furthermore, inthe exemplary embodiment, additional output channels may include, butnot be limited to, an operator interface monitor.

Further, as used herein, the terms “software” and “firmware” areinterchangeable, and include any computer program storage in memory forexecution by personal computers, workstations, clients, and servers.

As used herein, the term “non-transitory computer-readable media” isintended to be representative of any tangible computer-based deviceimplemented in any method or technology for short-term and long-termstorage of information, such as, computer-readable instructions, datastructures, program modules and sub-modules, or other data in anydevice. Therefore, the methods described herein may be encoded asexecutable instructions embodied in a tangible, non-transitory, computerreadable medium, including, without limitation, a storage device and amemory device. Such instructions, when executed by a processor, causethe processor to perform at least a portion of the methods describedherein. Moreover, as used herein, the term “non-transitorycomputer-readable media” includes all tangible, computer-readable media,including, without limitation, non-transitory computer storage devices,including, without limitation, volatile and nonvolatile media, andremovable and non-removable media such as a firmware, physical andvirtual storage, CD-ROMs, DVDs, and any other digital source such as anetwork or the Internet, as well as yet to be developed digital means,with the sole exception being a transitory, propagating signal.

As used herein, unless specified to the contrary, “modem terminationsystem,” or “MTS′” may refer to one or more of a CMTS, an opticalnetwork terminal (ONT), an optical line terminal (OLT), a networktermination unit, a satellite termination unit, and/or other terminationdevices and systems. Similarly, “modem” may refer to one or more of aCM, an optical network unit (ONU), a digital subscriber line (DSL)unit/modem, a satellite modem, etc.

As used herein, the term “database” may refer to either a body of data,a relational database management system (RDBMS), or to both, and mayinclude a collection of data including hierarchical databases,relational databases, flat file databases, object-relational databases,object oriented databases, and/or another structured collection ofrecords or data that is stored in a computer system.

Furthermore, as used herein, the term “real-time” refers to at least oneof the time of occurrence of the associated events, the time ofmeasurement and collection of predetermined data, the time for acomputing device (e.g., a processor) to process the data, and the timeof a system response to the events and the environment. In theembodiments described herein, these activities and events occursubstantially instantaneously.

The embodiments described herein provide innovative systems and methodsfor efficiently determining reflection and transmission coefficientssuch that, in the case of an impairment or interference on thetransmission line, signal correction may be advantageously performed atthe end user termination device, as well as elsewhere along thetransmission path. In an exemplary embodiment, full band capture (FBC)is performed by a network analyzer disposed at a fiber node of a wiredcommunication network. In some embodiments, FBC is performed at a hubMTS, or between the MTS and an end-user modem by a standalone dedicatednetwork analysis hardware unit, or virtually through a software moduleof a processor programmed for implementing network analysis of thespectral frequency transmitted along the transmission path.

FIG. 1 is a schematic illustration depicting a wired transmission system100. Transmission system 100 represents a cable transmissioninfrastructure or a hybrid fiber coaxial (HFC) network, and includes afiber node 102, which may be connected downstream of a communicationshub, optical line terminal (OLT), modem termination system (MTS), and/orcentral office (not shown in FIG. 1 ). Fiber node 102 is additionallyconnected to a data transmission line 104 for further transmission to aplurality of end users 106 in communication with data transmission line104 over a plurality of taps 108, respectively, disposed along a lengthof data transmission line 104. Data transmission line 104 is depicted inFIG. 1 as a coaxial cable for ease of explanation. End users 106represent downstream termination units, such as a subscriber device, amodem, a cable modem (CM), customer premises equipment (CPEs), abusiness user link, and/or an optical network unit (ONU). For simplicityof explanation, one end user 106 is depicted in FIG. 1 .

In operation of system 100, data transmission line 104 is damaged at adamage location 110 located between fiber node 102 and end user 106(between third and fourth taps 108, according to the example depicted inFIG. 1 ). A technician (not shown) runs a test on transmission line 104to detect damage location 110 using a test instrument 112 (including ahigh impedance probe 114), or a radar test set. A network analyzer 116measures a time domain response 118 of the test signal, which exhibits aparticular energy recursion 120 corresponding to the damage to datatransmission line at damage location 110, which is disposed adamage-to-analyzer distance L_(D) with respect to the disposition ofnetwork analyzer 116.

In additional operation of system 100, network analyzer 116 alsomeasures the reflection coefficient F, or gamma, of the test signal, andplots the measured reflection coefficient on a Smith Chart, as describedfurther below with respect to FIG. 2 . That is, the Smith Chart may beused here as a tool for visualizing the impedance of a transmission line(or antenna system) as a function of frequency.

FIG. 2 is a graphical illustration depicting a Smith Chart plot 200 ofenergy recursion 120, FIG. 1 . Plot 200 represents a combined impedancegrid (i.e., right-to-left) and admittance grid (i.e., left-to-right),and depicts a transmission coefficient 202 and a reflection coefficient204 of the measured energy recursion 120. However, as in the scenariodepicted in FIG. 1 , where the length L_(D) of the cable (i.e., portionof data transmission line 104) connected to network analyzer 116 is notknown, the nature of the measured reflection may not be determined fromthe frequency-based reflection coefficient 204 without performance of anadditional on-site calibration to remove the effects of the cable (e.g.,an on-site drop-cable test).

More particularly, as described above the F magnitude of reflectioncoefficient 204 may remain constant over a length of transmission line104, but the F phase of reflection coefficient 204 will change, that is,rotate or spin, according to the length of transmission line. On aconventional Smith Chart representation therefore, reflectioncoefficient 204 will be seen as a constant-radius rotation from a pointrepresenting a load value for F. This rotation, or spin, represents achange in phase of the complex number, and is a rotation at constantradius because the magnitude of the reflection coefficient remainsconstant. The longer the length of cable in question, the faster thespin on reflection coefficient 204. To compensate for this spin in aconventional system, sometimes referred to as “de-spinning F,” theon-site calibration must be performed at the terminating modem using aspecified, known length of drop cable (e.g., 100 ft) connected to themodem. As also described above, this process is resource intensive,requires the technician to be present at the site of the modem. Thischallenge is overcome according to the embodiments described furtherbelow.

FIG. 3 is a schematic illustration depicting an exemplary wiredtransmission system 300. Similar to transmission system 100, FIG. 1 ,exemplary system 300 includes a fiber node or communications hub 302connected to a data transmission line 304 for transmission to aplurality of end user termination devices 306 proximate or co-located atrespective customer premises 308. In the exemplary embodiment depictedin FIG. 3 , end user termination device is illustrated as a single modemfor ease of explanation. The person of ordinary skill in the art though,will understand that different, and/or additional termination devicesmay be implemented without departing from the scope of the embodimentsdescribed herein.

In an exemplary embodiment, modem 306 is connected to a respective tap310 disposed along data transmission line 304 (Tap4 310 ₄, in thisexample) downstream of a line impairment 312 (e.g., damage to a coaxialcable) from which at least one reflection signal arises. Similar tosystem 100, exemplary system 300 further includes a network analyzer 314having a processor and a memory (not separately shown) configured tomeasure or obtain the spectral frequency response of data transmissionline 304 (e.g., FIG. 5 , discussed further below).

In the exemplary embodiment illustrated in FIG. 3 , network analyzer 314is shown to be disposed within or co-located with hub/fiber node 302. Insome embodiments, network analyzer 314 may be disposed remotely from hubor fiber node 302 along data transmission line 304 between hub/node 302and modem 306. Optionally, network analyzer 314 may cooperate with anexternal testing device 316 (e.g., a high impedance probe connected to aspectrum analyzer or data acquisition device, such as a software definedradio (SDR) or analog-to-digital converter (ADC) in communication withnetwork analyzer 314) in contact with data transmission line 304.

Different though, from system 100, network analyzer 314 is furtherconfigured to advantageously utilize technology recently developed bythe present Assignee with respect to FBC proactive network maintenance(PNM) techniques implemented for Data Over Cable Service InterfaceSpecification (DOCSIS) v3.1, in which magnitude traces of the spectralfrequency band of the transmission are captured. Systems and methods forsuch innovative PNM techniques are described in greater detail inco-pending U.S. patent application Ser. No. 15/651,971, filed Apr. 12,2018, the subject matter and disclosure thereof which is incorporated byreference herein. The following embodiments illustrate furtherinnovations to modify this existing captured data to de-spin thereflection coefficient at the modem site without requiring a drop-cabletechnician on-site calibration.

In exemplary operation of system 300, network analyzer 314 captures thespectral data of transmission line 304 (e.g., FIG. 5 , below). In anexemplary embodiment network analyzer 314 functions as a measuringdevice for measuring the captured spectral frequency data, and also as aprocessing device for analyzing the captured spectral data to calculatea calibration correction at a remote location. In some embodiments, theactual measurement of the spectral data may be performed by a separatemeasuring device, and the data thereof communicated to network analyzer314. In at least one embodiment, data analysis may be performed remotelyby a third party processor in communication with at least one ofhub/node 302 and modem 306, such as in the case of a Cloud networkprocessing system or an Internet web services host.

In an embodiment, network analyzer 314 represents a spectrum analyzerand/or a data acquisition device directly or indirectly brought intocontact with transmission line 306 while operational. Theanalyzer/acquisition device thus is able to observe, for example,standing waves having a reflection on the transmission path (e.g.,arising from linear addition and subtraction of signals at differentfrequencies, caused by two signals traveling in opposite directionsalong transmission line 306).

Exemplary equalization systems and methods are described in greaterdetail in co-pending U.S. patent application Ser. No. 15/481,135, filedApr. 6, 2017, the subject matter and disclosure thereof which isincorporated by reference herein. U.S. U.S. patent application Ser. No.15/481,135, for example, discloses innovative techniques fortransforming complex frequency signals into the time domain by assumingthat the imaginary values for a transmitted cable signal are zero, withthe corresponding real values being measured signal magnitudes only.That is, in this co-pending application, the complex data points neednot be captured, and instead simply the real magnitude values. Thepresent disclosure expands on this time domain vs. frequency domainconcept through innovative techniques for analyzing F vs. Time, asopposed to the conventional techniques that only consider that is F withrespect to Frequency. This innovative concept is further described withrespect to FIGS. 4A and 4B, below.

The following embodiments are described below with respect to networksand systems for which spectral band capture is performed. The person ofordinary skill in the art will understand that these examples areprovided for illustrative purposes, and are not intended to be limiting.The present techniques may be applied to a number of other applications,including without limitation, wired communications (e.g., cable orfiber, wired MIMO), wireless (e.g., radar, wireless MIMO), recording,signal detection, and interfering signal rejection.

FIGS. 4A and 4B are graphical illustrations depicting a transmissionSmith Chart plot 400 and a reflection Smith Chart plot 402,respectively, for the signals measured by system 300, FIG. 3 . Moreparticularly, transmission Smith Chart plot 400 illustrates a simplifiedcomplex plot of scattering parameter S₂₁ (i.e., the forward transmissioncoefficient), and reflection Smith Chart plot 402 illustrates asimplified complex plot of scattering parameter S (i.e., the inputreflection coefficient). As can be seen from plot 402, the reflectedsignal has multiple reflections. On the other hand, as seen from plot400, the transmitted signal is dispersed (e.g., the resulting patternfrom signal path delays). However, when considering only the frequencydata of a conventional Smith Chart, a spin 404 of a particularcoefficient cannot be determined.

Accordingly, whereas a conventional Smith Chart only plots reflectioncoefficients against frequency, the present embodiments demonstrate howsuch complex responses may also be plotted with respect to time. In thisregard, it is helpful to visualize a time axis for plots 400, 402 asbeing perpendicular to the two-dimensional plane of the respective SmithChart, and increasing in time magnitude in the direction away from theviewer. A F-vs-time analysis may thus be achieved, for example, byperforming an inverse Fourier transform (IFT), or inverse fast Fouriertransform (IFFT) or inverse discrete Fourier transform (IDFT), on thecaptured spectral frequency data and plotting the inverse-transformeddata accordingly, and a cursor on a peak may show time delay, magnitude,and phase angle of the respective reflection. It may be noted that somesuch responses might be considered, at a distance, to be reactive,inductive, or capacitive.

In other words, a plot of the IFT of the network response (i.e., F vs.time) on a Smith Chart would render the center of the plot as indicatingno reflected signal from a given time window, while the left-mostboundary still represents a short and the right-most boundary stillrepresents an open (both from the given time window). As a cursor movesacross, or more aptly, into, the plot with increased time, the plottedresponse will move away from the middle of the traditionaltwo-dimensional Smith Chart plane in the direction of the impedancemismatch at the distance at issue. This expansion of the conventionalcoefficient parameters will not only greatly increase the ability of atechnician to troubleshoot a defective plant problem using reflectioncoefficients, but it will further avoid the need, in many cases, for thetechnician to have to calibrate the affected modems on-site. Thedispersion and associated angle for each signal path may thus be furthershown (e.g., three-dimensional plot of the transmission coefficientshown in plot 402, FIG. 4B), along with a relevant time of signalarrival.

Thus, at a high level, the present embodiments may be considered toeffectively utilize innovative principles for generating a “dynamicSmith Chart,” namely. a three-dimensional Smith Chart having time as anaxis perpendicular to the traditional Smith Chart plane (i.e., aSmith-Chart-over-time). As demonstrated by the techniques describedfurther below, the “dynamic Smith Chart” concept enables a calculationof a “spin coefficient”, and remotely from the terminal device itself(or at the terminal device, but without requiring a separate drop-cablecalibration procedure), which may be used to “de-spin” thetransmission/reflection coefficients plotted on conventional SmithCharts.

In other words, whereas conventional techniques enable the networkanalyzer to establish a calibration plane (e.g., a Smith Chart) forcalibration with a known reference (e.g., a drop cable), the presentcalibration techniques function to correct the calibration planeremotely, effectively extending conventional parameters by enabling thedetermination of both the distance from the reflection and the spincoefficient without requiring an on-site reference test by a technicianto stop the spin.

According to the present systems and methods, the terminal device ormodem may be easily instructed to make the calibration according to theremotely-calculated (or self-calculated) de-spinning spin coefficient.The relevant spectral measurements and de-spin calculations may be maderemotely from the modem, and the modem simply instructed to make thenecessary correction according to the remote measurements and/or spincalculations. In some embodiments, the modem may be advantageouslyprogrammed to automatically make such corrections based on the obtainedspectral data or FBC, thereby effectively rendering the modem its owntesting device.

In some embodiments, the reflection data may be obtained using a TimeDomain Reflectometer (TDR), which is a baseband instrument. At a givendistance, an open will deflect the trace therefrom upward, whereas ashort will deflect the trace downward. By looking at frequency domaindata as the measurement frequency approaches DC (i.e., 0 Hz), areflection coefficient may be estimated from the TDR observations. Withnetwork measurements, a DC response is often not possible as lowfrequency signals are not passed by the network. For example a typicaldownstream cable amplifier bandwidth is approximately 54 MHz at thelower end thereof. A waveguide is another example of a device which doesnot pass low frequencies. TDR measurements though, often require thatthe network be taken down, that is, go off-line or out-of-service, sothat a test signal may be accurately transmitted and measured along thetransmission path thereof. As described herein, the present techniquesmay be advantageously implemented for an in-service transmissionnetwork.

Thus, in an exemplary embodiment, in the case where a noise-like signal(such as a digital transmission) is combined with a delayed copy ofitself, the magnitude of the combined signals will add to or subtractfrom one another, depending on the particular frequency of overlap. Inthe case where the frequencies are exactly known, the reflectioncoefficient of the added signal may be determined by heterodyning thesignal down to DC (e.g., FIG. 6 , described further below), followed bya phase correction (e.g., FIG. 7 , described further below). For thisexemplary embodiment, the phase correction may be based on both theduration of the time delay, as well as the frequency offset. In theexemplary embodiment, these innovative techniques effectively convert anRF measurement into a baseband measurement, and thereby more accuratelyrevealing the reflection coefficient(s).

According to the systems and methods described herein, a technician,network analyzer, hub/node processor, and/or modem processor isadvantageously enabled to remotely (and also locally) determine thephase of a reflection from the calculated spin coefficient, andirrespective of whether an impairment to the transmission line is from ashort or an open, or whether inductive or capacitive. Indeed, using thespin coefficient calculation, a technician may be enabled to moreaccurately estimate the nature of the impairment prior to inspection, asopposed to the location only.

In an alternative embodiment, the phase of the reflection coefficient,and thus the spin coefficient, may be determined by extrapolating arotating-phasor of frequency response (e.g., FIG. 5 , below) back to DC,and then detecting whether the extrapolated rotating-phasor resulted at+90 degrees, — 90 degrees, or some other angle. In other words, thisalternative technique effectively follows the screw-like rotational pathdownward to the DC floor.

In accordance with the several techniques described herein, it isobserved that an increase in time delay will cause a phase rotation offrequency domain coefficients. In a similar manner, a change infrequency (e.g., heterodyning) will cause a phase rotation of the timedomain coefficients.

In a particular illustrative example, a measured linear channel responseis represented as F(n), and from a first frequency value f1 to a secondfrequency value f2. In this example, for ease of explanation, f1 is setto a zero value, and each additional time series coefficient, f[k], maybe calculated according to the following conventional IDFT equation:

$\begin{matrix}{{f\lbrack k\rbrack} = {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}{{F\lbrack n\rbrack}e^{{+ j}\frac{2\pi}{N}{nk}}}}}} & \left( {{Eq}.1} \right)\end{matrix}$

Thus, in the case where f1 is zero, the equation yields the correctanswer. However, in the case where f1 is not 0 Hz, each time sample f[k]will have a phase error, per time sample, equal to 2π*SC/N radians,where SC is an integer index number of subcarriers needed to adjustf1[0] to 0 Hz, and N is the FFT size.

In a specific illustrative example, network analyzer 314 obtainsavailable spectrum data for an OFDMA channel having 720 subcarrierpredistortion coefficients available every 50 KHz, e.g., from 5.75 MHzto 41.75 MHz. Accordingly, the “0 Hz” coefficient for f1, if it wereavailable, would be is 115 frequency domain samples below 5.75 MHz.Therefore, beginning at 5.75 MHz, and using an FFT size of 512, thephase correction for each subcarrier, per time sample, would be2π*SC/512 radians. Thus, to get back to 0 Hz, the integer index numberSC would be 115 (i.e., SC=215) if beginning the first sample, but 215 ifbeginning on the 100^(th) sample (i.e., SC=215).

Exemplary processing steps for obtaining the spin coefficient aredescribed further below with respect to FIGS. 5-8 . In an exemplaryembodiment, these processing steps may be implemented with respect totransmission system 300, FIG. 3 , one or more of the several elementstherein, a transmission system implementing a DOCSIS protocol, or adigital transmission system implementing a non-DOCSIS OFDM or OFDMAprotocol.

In exemplary operation, processing steps to determine a spin coefficientmay include one or more of: (a) measuring a complex signal level againstfrequency; (b) adjusting the start frequency of the measured signal toDC (e.g., 0 Hz), such as by way of digital signal processing (DSP)and/or heterodyning; (c) perform an IFT/IFFT on the measured signal toconvert frequency domain samples into time domain samples; (d) correctthe phase on each converted time domain sample using time value(s)and/or frequency shift(s); and (e) determine a corrected reflectioncoefficient for each corrected time domain sample to de-spin thereflection coefficient. Optionally, an additional processing step mayconvert the corrected complex time domain sequence (i.e., real andimaginary values) to a real-value-only time domain sequence forincreased resolution and accuracy. For example, to obtain a moreaccurate phase value for the first such time domain sample, beginningagain at subcarrier 115 (i.e., SC=115) 1.411 radians may be added tothis first time domain sample, 2.822 radians to the second time domainsample, and so forth.

In an exemplary embodiment, the present terminal device/modem correctiontechniques may be accomplished in coordination with computer code orexecutable instructions, for example, such as that as shown below withrespect to the executable code listed in the attached Computer ProgramListing. Additional embodiments, variations, and real-worldproof-of-concept simulation results of these exemplary processing stepsare described further below with respect to FIGS. 5-7 .

FIG. 5 is a graphical illustration depicting a comparative plot 500 ofsimulated frequency spectral data captured by network analyzer 314, FIG.3 . More particularly, comparative plot 500 includes a first subplot 502(Series1, solid line) and a second subplot 504 (Series2, dashed line),respectively illustrating test results of 512 sampled I (real) and Q(imaginary) complex spectral data points selected from a captured1024-point frequency domain simulated sequence set of 1024 points,beginning at 0 Hz. That is, the phases are correct at 0 Hz; however, asdescribed above, a selected subset starting point may be selected fromany random frequency between 0 and 512, which is particularly valuablefor networks that do no operate below 54 MHz. For example, a startfrequency could be 100 MHz, and the end frequency would then be 612 MHz.In the exemplary test case shown in FIG. 5 , a real echo of 20% wasadded at t=8 samples, and an imaginary echo of 20% was added at t=16samples.

In the exemplary embodiment, the spectral data samples are measured(i.e., sampled) by network analyzer co-located with the hub or fibernode. However, as described above, in other embodiments, networkanalyzer 314 need only obtain the spectral data from a sampling unitcontacting the transmission line at some testing point from thetermination unit/modem to the hub or node. The sampling unit may beintegral with network analyzer 314, or may be a separate device disposedremotely from network, analyzer 314. In at least one embodiment, thetesting point may be proximate the modem. Nevertheless, the presentembodiments advantageously remove the need to co-locate the testingpoint (e.g., the sampling unit) with the correction point (e.g., themodem). In some instances, the sampled spectral data may be obtainedfrom a device already dedicated to measure and sample the frequencyspectrum of the transmission line, such as a DOCSIS 3.1 FBC PNM unit.

The versatility of this configuration for spectral capture thus enablesthe present techniques to be easily implemented or coordinated withexisting fiber node and/or wired transmission architectures with minimaladditional hardware and programming costs. System 300 may thus furtheradvantageously function as a standalone system, or may be fullybackwards-compatible with existing modem, fiber node, and MTS systemspresently in use.

The present coefficient expansion techniques of system 300 are furtheralso fully compatible with the extended spectrum of DOCSIS 3.1, andparticularly in the case where it is desirable to perform datacollection and/or analysis at a different location from where thecalibration/correction occurs (e.g., the modem). Such distributedcalibration/correction techniques may be particularly desirable wherethe cost-benefit weigh in favor of collecting the data at one location(e.g., network analyzer, modem, or a testing instrument, such as a TDR),analyzing the sampled data at another location (e.g., network analyzer,modem, or Cloud/Internet-based web services), and then performing thenecessary correction calibration at the target termination device/modem.The present correction/calibration techniques are thus furtheradvantageous in the case where both the data analysis and correction areperformed at the modem (assuming sufficient programming and processingpower of modem processor 318, FIG. 3 ), in that spectral capture andsampling may be easily performed at a central location without requiringtechnician travel to the modem location and perform a drop-cable test.

FIG. 6 is a graphical illustration depicting a comparative time domainplot 600 of comparative frequency plot 500, FIG. 5 . More particularly,comparative time domain plot 600 includes a first subplot 602 (solidline) and a second subplot 604 (dashed line) corresponding to firstsubplot 502 and second subplot 504, FIG. 5 , respectively, but afterperformance of an IFFT operation on the respective spectral datathereof. That is, subplots 602, 604 represent the time domaincounterparts of the frequency domain samples of subplots 502, 504,respectively. In a conventional transmission system, the IFFT processoris typically disposed with the test instrument (e.g., probe, networkanalyzer, etc.); however, according to the present embodiments, suchoperations may be distributed (e.g., spectral measurement by a probe oranalyzer, IFFT and other DSP by a Cloud or web service, and thecorrection by the modem). In the exemplary embodiment, samplingoperations are performed at the testing point.

For the real-world results depicted in FIG. 6 , the time domain samplesshow t=8 at 0 degrees, and t=16 at 90 degrees, as indicated by therespective peaks of subplots 602, 604. As can also be seen from timedomain plot 600, the amplitudes of both subplots are correct, but due tothe frequency shift, the respective phase data on the delayed timedomain samples are not. That is, in the case where there is no frequencyshift (e.g., beginning at 9 Hz), the phase data is correct, and thus thephase angles would not need correction.

Nevertheless, real-world transmission systems do not all start at 0.Accordingly, an additional phase correction step is illustrated furtherbelow with respect to FIG. 7 .

FIG. 7 is a graphical illustration depicting a phase correction plot 700of comparative time domain plot 600, FIG. 6 . More particularly, phasecorrection plot 700 includes a first subplot 702 (solid line) and asecond subplot 704 (dashed line) corresponding to first subplot 602 andsecond subplot 604, respectively, of comparative time domain plot 600,FIG. 6 , but after performance of a phase correction operation on therespective time domain data samples thereof. That is, subplots 702, 704represent the respective time domain data of subplots 602, 604,respectively, after a phase rotation correction. As can be seen from thecorrected time data illustrated in FIG. 7 , the phase correctionadvantageously follows a linear relationship with both the frequencyoffset and time index.

According to the embodiments described above, in the case of damage to atransmission line, the phase correction to “de-spin” the resultingreflection coefficient(s) may be determined, other than the physicalcapture of spectral data from the transmission system by entirelymathematical algorithms programmed into one or more of the respectivecomponent processors typically found in existing digital transmissionsystems. Such new and additional algorithmic capabilities, however, arehardly abstract; some spectral measurement is required, and the terminaldevice or modem is configured to receive instructions to utilize thecalculated “spin coefficient” to de-spin the rotating reflectioncoefficient at the modem. Accordingly, whether the analysis of thespectral data and calculation of the spin coefficient are centralized toa single system component, or distributed among several components ofthe transmission system, the present systems and methods avoid the needfor an on-site technician drop-cable test in many cases.

FIG. 8 is a graphical illustration depicting a conversion plot 800 ofphase correction plot 700, FIG. 7 . More particularly, in an optionalsubprocessing step, the complex (i.e., real and imaginary) data fromphase correction plot 700 is further converted into first subplot 802and second subplot 804 of real-only coefficients. That is, real firstsub-plot 802 is effectively doubled, and imaginary second subplot iseffectively reduced to zero. In an exemplary embodiment, the real-onlyconversion is accomplished by first creating a complex conjugate mirrorin the frequency domain, and then performing an IFFT operation on thecomplex conjugate mirror. Using this optional processing technique,twice the number of frequency points and time samples may be obtained,thus greatly improving the resolution and accuracy of the calculatedspin coefficient and phase correction at the modem. As may be noted fromsecond subplot 804, at t=16, the imaginary response resembles a Hilberttransform. This additional computational processing step thus providesstill further significant advantages over conventional reflectionevaluation techniques.

The present systems and methods are therefore particularly useful in thecase where the delay in a reflection or echo is relatively short. Insuch cases, the corresponding separation between ripple peaks in thefrequency domain will be large. However, due to windowing effects, thepresence of a significant DC term in the time series (i.e., the x[0]complex term) of the corresponding time sequence may render anreflection having a smaller, more slightly delayed, echo more difficultto resolve. Accordingly, the corresponding convolution is a sin(x)/xseries, as opposed to a Kronecker delta function. It may desirable, insuch scenarios, to remove the DC term to simplify the correctioncalculations, particularly in the case where a window function isdesired to reduce ringing. Removal of the DC term may also beadditionally advantageous in the case where the DC term is spread, suchas may result from group delay or tilt.

Although the present techniques provide significant advantages to manydifferent types of transmission systems, the systems and methodsdescribed herein are particularly applicable to cable transmissionnetworks. For example, cable network technicians are known to commonlyutilize TDR techniques to obtain a magnitude-only, DC-referenced,real-valued impulse response derived from a frequency domain networkanalyzer test or FBC. According to the present techniques though, theseexisting conventional measurement techniques may be readily adapted, asdescribed herein, to further determine whether the obtained impulseresponses indicate an open, a short, an inductive impairment, acapacitive impairment, or any complex vector.

The present systems and methods are of further advantageousapplicability to a wide variety of complex data set where frequency(ies)of test points is known. The versatility of the present embodiments, forexample, enable use with not only upstream predistortion OFDMA data, butalso with complex data sets such as those implemented as ATDMApredistortion coefficients. Furthermore, the innovative techniquesdescribed herein are also applicable even in the case where a real-onlyfrequency domain sequence is available (e.g., a Real-Only NetworkAnalyzer or TDR). In such cases, a complex sequence may be produced fromthe real-only sequence, and the techniques described above implementedwith respect to the resulting complex sequence. Thus, the wider thebandwidth, the better the resolution on lower-delay echoes, as well asmultiple echoes that are close to one another in frequency.

The systems and methods described herein are therefore of particularusefulness with respect to the many millions of DOCSIS 3.1 modems andcable modems presently deployed throughout the world. According to thepresent embodiments, many of these millions of deployed modems, in theevent of damage to a connected coaxial cable, may be advantageouslyconfigured to self-calibrate without the need for a techniciandrop-cable on-site measurement. The present embodiments though, are alsoparticularly advantageous with respect to OFDM signals that are notnecessarily transmitted according to DOCSIS 3.1. The embodiments hereinthus represent significant improvements to one or more of thetransmission system, the network analyzer, and the terminaldevice/modem.

Although specific features of various embodiments may be shown in somedrawings and not in others, such is for convenience only. In accordancewith the principles of the systems and methods described herein, anyfeature of a drawing may be referenced or claimed in combination withany feature of any other drawing.

Some embodiments involve the use of one or more electronic or computingdevices. Such devices typically include a processor, processing device,or controller, such as a general purpose central processing unit (CPU),a graphics processing unit (GPU), a microcontroller, a reducedinstruction set computer (RISC) processor, an application specificintegrated circuit (ASIC), a programmable logic circuit (PLC), aprogrammable logic unit (PLU), a field programmable gate array (FPGA), adigital signal processing (DSP) device, and/or any other circuit orprocessing device capable of executing the functions described herein.The methods described herein may be encoded as executable instructionsembodied in a computer readable medium, including, without limitation, astorage device and/or a memory device. Such instructions, when executedby a processing device, cause the processing device to perform at leasta portion of the methods described herein. The above examples areexemplary only, and thus are not intended to limit in any way thedefinition and/or meaning of the term processor and processing device.

This written description uses examples to disclose the embodiments,including the best mode, and also to enable any person skilled in theart to practice the embodiments, including making and using any devicesor systems and performing any incorporated methods. The patentable scopeof the disclosure is defined by the claims, and may include otherexamples that occur to those skilled in the art. Such other examples areintended to be within the scope of the claims if they have structuralelements that do not differ from the literal language of the claims, orif they include equivalent structural elements with insubstantialdifferences from the literal language of the claims.

Computer Program Listing //RealOnlyfromFD C++ code #include <iostream>#include <math.h> #include <stdio.h> #include <stdlib.h> #include<malloc.h> #include <stdlib.h> #include <string.h> typedef struct {floatreal, imag;} COMPLEX; extern void fourier(COMPLEX *,int);//fast Fouriertransform extern void ifourier(COMPLEX *,int);//fast inverse Fouriertransform extern void black(COMPLEX *,int); extern void makeSignal( );float vr[1024],vi[1024], Pl=3.14156,dB; int main( ) {   makeSignal( );  int i,j,k=32;   char zz;   float ang, ang2[512], mag, scale, cang[512],cr1[1024],ci1[1024],cr2[1024],ci2[1024],cr3[1024],ci3[1024],cr4[1024],ci4[1024];  FILE *inputdata, *outputdata;   COMPLEX *y;   y=(COMPLEX*)calloc(1024, sizeof(COMPLEX));   if(!y){printf(″\n Unable to allocateinput memory.\n″);printf(″\x7″);exit(1);};   COMPLEX *x;   x=(COMPLEX*)calloc(1024, sizeof(COMPLEX));   if(!x){printf(″\n Unable to allocateinput memory.\n″);printf(″\x7″);exit(1);};   COMPLEX *xx;  xx=(COMPLEX*) calloc(1024, sizeof(COMPLEX));   if(!xx){printf(″\n Unableto allocate input memory.\n″);printf(″\x7″);exit(1);};   if( (inputdata= fopen(″input1.txt″, ″r″) ) == NULL){    printf(″could not openfile.\n″);    system(″pause″);    exit(0);   }   if( (outputdata =fopen(″output1.txt″, ″w″) ) == NULL){    printf(″could not openfile.\n″);    system(″pause″);    exit(0);   }   for(i=0;i<1024;i++){  fscanf(inputdata,″%f%f%f%f%f%f%f%f″,&cri[i],&ci1[i],&cr2[i],&ci2[i],&cr3[i],&ci3[i],&cr4[i],&ci4[i]); ]);   // if(i<10)printf(″%d\t%f\t%f\n″,i,cr1[i],ci1[i]);   }   int offset =120;//  int startNr =115+offset; //7.75/.05 MHz   int startNr = 0+offset;//for makeSignal data, which starts at 0 MHz  fprintf(outputdata,″Offset = %d\n″,offset);   for(i=0;i<512;i++){   // y[i].real = cr3[i+offset];//10.75 Mhz to 36.35 MHz   //  y[i].imag =ci3[i+offset];    y[i].real = vr[i+offset ];    y[i].imag =vi[i+offset];   //printf(″y %d\t%f\t%f\n″,i,y[i].real,y[i].imag);   }  //black(y,512);//window data   ang = − atan2(y[0].imag,y[0].real);  printf(″ang=%f radians\n″,ang); ang = 0;  //https://en.wikipedia.org/wiki/Rotation_matrix  for(i=0;i<512;i++){//rotate to get xx[0] to be 0    //xx[i].real =y[i].real * cos(ang) − y[i].imag *sin(ang);    //xx[i].imag =y[i].real * sin(ang) + y[i].imag* cos(ang);    xx[i].real = y[i].real;   xx[i].imag = y[i].imag;   }   for(1=0;i<512;i++){   mag =sqrt(xx[i].real*xx[i].real + xx[i].imag*xx[i].imag);   ang =atan2(xx[i].imag,xx[i].real);   fprintf(outputdata,″f%d\t%f\t%f\t%f\t%f\n″,i,xx[i].real,xx[i].imag,mag,ang);   }  ifourier(xx,9);   for(i=0;i<512;i++){   mag =sqrt(xx[i].real*xx[i].real + xx[i].imag*xx[i].imag);   ang =atan2(xx[i].imag,xx[i].real);   fprintf(outputdata,″t%d\t%f\t%f\t%f\t%f\n″,i,xx[i].real,xx[i].imag,mag,ang);   }   //nowcorrect for angle in TD due to freq mixing   for(i=0;i<512;i++){   ang2[i] = ((float)startNr*(float)i*2*Pl)/512;   // printf(″%d\t%f\n″,i,ang2[i]); } // now correct phasefor(i=0;i<512;i++){    x[i].real = xx[i].real * cos(ang2[i]) −xx[i].imag *sin(ang2[i]);    x[i].imag = xx[i].real * sin(ang2[i]) +xx[i].imag* cos(ang2[i]);    mag = sqrt(x[i].real*x[i].real +x[i].imag*x[i].imag);    ang = atan2(x[i].imag,x[i].real);// printf(″~%d\t%f\t%f\t%f\t%f\n″,i,x[i].real,x[i].imag,mag,ang);  fprintf(outputdata,″~%d\t%f\t%f\t%f\t%f\n″,i,x[i].real,x[i].imag,mag,ang);} fourier(x,9);//back into fd to make a mirror   for(i=1;i<512;i++){   x[1024-i].real =x[i].real;    x[1024-i].imag = −x[i].imag;   }  x[512].real = x[0].real;   x[512].imag = x[0].imag;//should be zero  for(i=0;i<1024;i++){   fprintf(outputdata,″!f%d\t%f\t%f\n″,i,x[i].real,x[i].imag);   } ifourier(x,10);  for(i=1024-32;i<1024;i++){   fprintf(outputdata,″t%d\t%f\t%f\n″,i,x[i].real,x[i].imag);   }   for(i=0;i<1024;i++){  fprintf(outputdata,″t %d\t%f\t%f\n″,i,x[i].real,x[i].imag);   }  printf(″Hello″);   fclose(inputdata);   fclose(outputdata); }//end ofmain void makeSignal( ){   int i;   float a=.2, b=.2, n=16;   FILE*signal;    if( (signal = fopen(″signal.txt″, ″w″) ) == NULL){   printf(″could not open file.\n″);    system(″pause″);    exit(0);   }  float th = −Pl/2;   //float th = Pl;   for(i=0;i<1024;i++){    //vr[i] =1 + a*cos(1*Pl*n*(float)i/512) + b*cos(2*Pl*n*(float)i/512);    //vi[i]= − a*sin(1*Pl*n*(float)i/512) − b*sin(2*Pl*n*(float)i/512);    vr[i] =1 + a*cos(1*Pl*n*(float)i/512) + b*cos(th+(2*Pl*n*(float)i/512));   vi[i] = − a*sin(1*Pl*n*(float)i/512) − b*sin(th+(2*Pl*n*(float)i/512) );//  vi[i] = −vi[i];   fprintf(signal,″%d\t%f\t%f\t%f\n″,i,vr[i],vi[i],atan2(vi[i],yr[i]−1));   }    printf(″done making signal\n″); }//----------------------------------------------------------------------------------------------------------void fourier(COMPLEX *x,int m)//takes a fourier transform of size 2 tothe m power {   static COMPLEX *w;   static int mstore = 0;   static intn = 1;   COMPLEX u,temp,tm;   COMPLEX *xi,*xip,*xj,*wptr;   inti,j,k,l,le,windex;   doublearg,w_real,w_imag,wrecur_real,wrecur_imag,wtemp_real;   if(m != mstore){   if(mstore != 0) free(w);   mstore = m;   if(m == 0) exit(1);   n =1<< m;   le =n/2;   w = (COMPLEX *) calloc(le−1,sizeof(COMPLEX));  if(!w) {   printf(″\nUnable to allocate array\n″);   exit(1);   }   arg= 4.0*atan(1.0)/le;   wrecur_real = w_real = cos(arg);   wrecur_imag =w_imag = −sin(arg);   xj = w;   for (j = 1; j < le ; j++) {    xj->real= (float)wrecur_real;    xj->imag = (float)wrecur_imag;    xj++;   wtemp_real = wrecur_real*w_real − wrecur_imag*w_imag;    wrecur_imag =wrecur_real*w_imag + wrecur_imag*w_real;    wrecur_real = wtemp_real;  }   }   le = n;   windex = 1;   for (l = 0 ; l < m ; l++) {   le = le/2;  for(i = 0; i < n; i = i + 2*le){   xi =x + i;   xip =xi + le;  temp.real = xi->real + xip->real;   tenip.imag = xi->imag + xip->imag;  xip->real = xi->real − xip->real;   xip->imag = xi->imag − xip->imag;  *xi = temp;   }   wptr = w + windex − 1;   for (j = 1; j < le ; j++) {  u = *wptr;   for(i = j; i < n ; i = i + 2*le){   xi = x + i;   xip =xi + le;   temp.real = xi->real + xip->real;   tenip.imag = xi->imag +xip->imag;   tm.real = xi->real − xip->real;   tm.imag =xi->imag −xip->imag;   xip->real = tm.real*u.real − tm.imag*u.imag;   xip->imag =tm.real*u.imag + tm.imag*u.real;   *xi = temp;   }   wptr = wptr +windex;   }   windex = 2*windex;   }   j = 0;   for (i = 1; i < (n−1);i++) {   k = n/2;   while(k <= j) {   j = j − k;   k = k/2;   }   j =j + k;   if (i < i) {   xi = x + i;   xj = x + j;   temp = *xj;   *xj=*xi;   *xi = temp;   } } }//end of fourier void ifourier(COMPLEX *x,intm) {   static COMPLEX *w;   /* used to store the w complex array */  static int mstore = 0;   /* stores m for future reference */   staticint n = 1;   /* length of ifft stored for future */   COMPLEX u,temp,tm;  COMPLEX *xi,*xip,*xj,*wptr;   int i,j,k,l,le,windex;   doublearg,w_real,w_imag,wrecur_real,wrecur_imag,wtemp_real;   float scale;  if(m != mstore) { /* free previously allocated storage and set new m */   if(mstore !=0) free(w);    mstore = m;    if(m == 0) exit(1);   /* ifm=0 then done */ /* n = 2**m = inverse fft length */    n =1 << m;    le= n/2; /* allocate the storage for w */    w = (COMPLEX *)calloc(le−1,sizeof(COMPLEX));    if(!w) {     printf(″\nUnable toallocate complex W array\n″);     exit(1);    } /* calculate the wvalues recursively */    arg = 4.0*atan(1.0)/le;   /* Pl/le calculation*/    wrecur_real = w_real = cos(arg);    wrecur_imag = w_imag =sin(arg); /* opposite sign from fft */    xj = w;    for (j = 1 ; j < le; j++){     xj->real = (float)wrecur_real;     xj->imag =(float)wrecur_imag;     xj++;     wtemp_real = wrecur_real*w_real −wrecur_imag*w_imag;     wrecur_imag = wrecur_real*w_imag +wrecur_imag*w_real;     wrecur_real = wtemp_real;    }   } /* startinverse fft */   le =n;   windex =1;   for (l =0; l < m ; l++){    le =le/2; /* first iteration with no multiplies */    for(i = 0; i < n ; i =i + 2*le){     xi = x +i ;     xip = xi + le;     temp.real = xi->real +xip->real;     temp.imag = xi->imag + xip->imag;     xip->real =xi->real − xip->real;     xip->imag = xi->imag − xip->imag;      *xi =temp;    } /* remaining iterations use stored w */    wptr =w + windex −1;    for (j = 1; j < le; j++) {    u = *wptr;    for( i = j ; i < n ; i= i + 2*le){     xi = x + i;     xip = xi + le;     temp.real =xi->real + xip->real;     temp.imag = xi->imag + xip->imag;     tm.real= xi->real − xip->real;     tm.imag = xi->imag − xip->imag;    xip->real = tm.real*u.real − tm.imag*u.imag;     xip->imag =tm.real*u.imag + tm.imag*u.real;     *xi =temp;    }    wptr = wptr +windex;   }   windex = 2*windex;   } /* rearrange data by bit reversing*/   j = 0;   for (i =1; i <(n−1); i++) {   k = n/2;   while(k <= j) {   j = j − k;    k = k/2;    }    j = j + k;    if (i < i) {     xi =x + i;     xj = x + j;     temp = *xj;     *xj = *xi;     *xi = temp;   }   } /* scale all results by 1/n */   scale = (float)(1.0/n);   for(i =0; i < n ; i++) {    x->real = scale*x->real;    x->imag =scale*x->imag;    x++;   } } void black( COMPLEX *x, int n) {   int i;  double black,factor;   factor = 8.0*atan(1.0)/(n−1);   for (i=0; i<n;++i){   black = 0.42 − 0.5*cos(factor*i) + 0.08*cos(2*factor*i);  x->real *= black;   x->imag *= black;   x++;   } }

The invention claimed is:
 1. A method of calibrating a terminal deviceconnected to a transmission line containing at least one impairment,comprising the steps of: obtaining a sequence of frequency domainsamples for a digital signal transmitted to the terminal device over thetransmission line; determining, from the obtained sequence of frequencydomain samples, a reflection coefficient from a frequency domainreflection signal of the digital signal arising from the at least oneimpairment; converting the sequence of frequency domain samples and thefrequency domain reflection signal into a time domain to generate acomplex time domain sample sequence having a real I time component andan imaginary Q time component; correcting the complex time domain samplesequence into a corrected time sequence having a phase value of theimaginary Q time component corresponding to a phase value of the real Itime component; calculating a correcting spin coefficient from thecorrected time sequence; and calibrating the terminal device with thecorrecting spin coefficient to mitigate a rotation of the reflectioncoefficient.
 2. The method of claim 1, wherein the step of obtainingcomprises a full band capture (FBC) of a frequency spectrum of thedigital signal and the frequency domain reflection signal by a testinginstrument connected to the transmission line.
 3. The method of claim 2,wherein the testing instrument is a network analyzer disposed upstreamof the terminal device.
 4. The method of claim 3, wherein the networkanalyzer is disposed proximate at least one of a hub, a central office,and a fiber node of a transmission network connected to the transmissionline.
 5. The method of claim 4, wherein at least one of the steps ofdetermining, converting, correcting, and calculating are executed by anode processor of the fiber node.
 6. The method of claim 5, wherein thestep of calibrating includes an instruction, from the node processor tothe terminal device, for the terminal device to execute a calibrationoperation according to the calculated correcting spin coefficient. 7.The method of claim 3, wherein at least one of the steps of determining,converting, correcting, and calculating are executed by the networkanalyzer.
 8. The method of claim 7, wherein the step of calibratingincludes an instruction, from the network analyzer to the terminaldevice, for the terminal device to execute a calibration operationaccording to the calculated correcting spin coefficient.
 9. The methodof claim 2, wherein the testing instrument is a high impedance probe incooperation with a data acquisition device disposed upstream of theterminal device.
 10. The method of claim 9, wherein the data acquisitiondevice includes a sampler, and wherein at least one of the steps ofdetermining, converting, and correcting are executed by the sampler. 11.The method of claim 2, wherein the testing instrument is the terminaldevice.
 12. The method of claim 11, wherein at least one of the steps ofdetermining, converting, correcting, and calculating are executed bythird party computational service in operable communication with theterminal device.
 13. The method of claim 11, wherein the terminal deviceis configured to self-calibrate according to the calculated correctingspin coefficient.
 14. The method of claim 1, wherein the terminal deviceis at least one of a modem, a cable modem (CM), an optical network unit(ONU), a digital subscriber line (DSL) unit, and a satellite modem. 15.A network analyzer coupled with a transmission line of a digitalcommunication network, the network analyzer comprising: a processor inoperable communication with the transmission line; and a memory devicefor storing computer-executable instructions, which, when executed bythe processor, cause the processor to: obtain a sequence of frequencydomain samples for a digital signal transmitted over the transmissionline to a downstream terminal device; determine, from the obtainedsequence of frequency domain samples, a reflection coefficient of afrequency domain reflection signal arising from a physical impairment ofthe transmission line disposed between the network analyzer and thedownstream terminal device; convert the sequence of frequency domainsamples and the frequency domain reflection signal into a time domain togenerate a complex time domain sample sequence having a real I componentand an imaginary Q component in the time domain; correct the complextime domain sample sequence into a corrected time sequence having aphase value of the imaginary Q time component corresponding to a phasevalue of the real I time component; calculate a correcting spincoefficient from the corrected time sequence; and instruct thedownstream terminal device calibrate a rotation of the reflectioncoefficient arising from the physical impairment using the calculatedcorrecting spin coefficient.
 16. The network analyzer of claim 15,disposed within a fiber node of the digital communication networkcontacting the transmission line.
 17. The network analyzer of claim 15,wherein the instructions further cause the processor to detect, based onthe calculated correcting spin coefficient, that the physical impairmentincludes at least one of an open circuit, a short circuit, an inductiveimpairment, and a capacitive impairment.
 18. The network analyzer ofclaim 15, further comprising a sampling unit, and wherein theinstructions further cause the processor to obtain the sequence offrequency domain samples for the digital signal directly from thesampling unit.
 19. The network analyzer of claim 18, further comprisinga measurement device in direct contact with the transmission line, andwherein the instructions further cause the processor to direct thesampling unit to sample a frequency spectrum of the digital signaldirectly measured by the measurement device.
 20. The network analyzer ofclaim 19, wherein the measurement device is disposed remotely from theprocessor.